If I asked you which of the following
figures is more symmetrical, I think you would intuitively realize that the one in
the upper left hand corner is least symmetrical, the rectangle is more symmetrical,
the triangle more symmetrical still and so on. This intuitive notion is consistent
with both definitions. If you think in terms of balance, the rectangle is balanced
about about both lines through the middle of the sides. This notion of balance has
an ``operational'' counterpart in the context of Weyl's definition. I can reflect
the rectangle through both lines through the center (by flipping it over) and it looks
exactly the same. This operation is a symmetry. The triangle clearly has three lines,
or axes, about which it is balanced. I can reflect it through any of these three lines
and it looks the same. The square has four symmetry axes, a pentagon has five, etc. In
this way, the statement that the pentagon is the most symmetrical object on the slide
takes on a precise meaning, but one that is consistent with the Oxford dictionary's
definition.
Hopefully it is also clear that each time you add a side of equal length, you add a
symmetry. If you keep the process up ad infinitum you get the most symmetrical two
dimensional shape conceivable...